Exact nonadiabatic part of the Kohn-Sham potential and its fluidic approximation

M. T. Entwistle and R. W. Godby

Phys. Rev. Materials 4, 035002 – Published 16 March 2020

Abstract

We present a simple geometrical “fluidic” approximation to the nonadiabatic part of the Kohn-Sham potential, $v_{KS}$ , of time-dependent density-functional theory (DFT). This part of $v_{KS}$ is often crucial, but most practical functionals utilize an adiabatic approach based on ground-state DFT, limiting their accuracy in many situations. For a variety of model systems, we calculate the exact time-dependent electron density and find that the fluidic approximation corrects a large part of the error arising from the “exact adiabatic” approach, even when the system is evolving far from adiabatically.

Received 15 November 2019
Accepted 25 February 2020

DOI:https://doi.org/10.1103/PhysRevMaterials.4.035002

Physics Subject Headings (PhySH)

Density functional calculations Density functional theory First-principles calculations Time-dependent DFT

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

M. T. Entwistle and R. W. Godby

Department of Physics, University of York, and European Theoretical Spectroscopy Facility, Heslington, York YO10 5DD, United Kingdom

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Issue

Vol. 4, Iss. 3 — March 2020

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Images

Figure 1
System 1: Two interacting electrons in a Gaussian potential well, with a uniform electric field applied at $t = 0$ , driving the electrons to the right and inducing a current. (a) The ground-state external potential (dashed purple) and exact ground-state electron density (dashed blue), along with the perturbed external potential (solid purple) and exact time-dependent electron density at $t = 8$ a.u. (solid blue). (b) The change in the exact electron density $[δ n (x, t) = n (x, t) - n (x, 0)]$ at $t = 8$ a.u. (short-dashed green), along with that obtained when using the exact $v_{KS}^{A}$ (solid blue), and when adding the exact $v_{KS}^{A}$ with the fluidic approximation $Δ A_{KS} = - u$ (dashed red). The exact adiabatic potential is clearly inadequate, but its error is substantially reduced by the fluidic approximation.
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Figure 2
The nonadiabatic correction to the KS potentials for system 1. (a) The exact $Δ A_{KS}$ (short-dashed green) and that obtained using the fluidic approximation $Δ A_{KS} = - u$ (dashed red), at $t = 8$ a.u. (b) The corresponding exact (short-dashed green) and fluidic (dashed red) $Δ v_{KS}$ in its scalar form. The fluidic approximation performs very well, even though the velocity field is non-uniform in both space and time. (The exact adiabatic approximation, of course, amounts to setting $Δ A_{KS} = Δ v_{KS} = 0$ .)
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Figure 3
System 2C: Three interacting electrons in an atomiclike potential, with a static sinusoidal perturbation applied at $t = 0$ , pushing the electrons apart. (a) The ground-state external potential (dashed purple) and exact ground-state electron density (dashed blue), along with the perturbed external potential (solid purple) and exact time-dependent electron density at $t = 5$ a.u. (short-dashed blue) and $t = 18$ a.u. (solid blue). (b) The change in the exact electron density at $t = 5$ a.u. (short-dashed green), along with that obtained when using the exact $v_{KS}^{A}$ (solid blue), and when adding the exact $v_{KS}^{A}$ with the fluidic approximation (dashed red). Even though the density is strongly disrupted, the fluidic approximation remains successful. (c) The same as (b) but at $t = 18$ a.u., where the dynamic xc contribution is very significant, evident by the completely inadequate result obtained with the fixed $v_{xc}$ (short-dashed gray) (see main text). Here, the exact $v_{KS}^{A}$ is better, but adding the fluidic approximation improves it further.
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Figure 4
System 3: Two interacting electrons in a tunneling system. Inset: The exact total electron number on the left-hand side ( $x < 0$ ) (short-dashed green); also the exact adiabatic (solid blue) and fluidic approximation (dashed red).
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